I've got a question in this problem. I also will talk about the definitions I'm using (I'm using a Ring as an abelian aditive group with a multiplicative semigroup and distributive laws) I have a commutative ring $R$ without zero divisors. The first question. I can make a Partial Ring of Quotients of this ring. Can't I make a Field of Quotients of this ring? Why is the unity needed?
The second question is about what I was doing. I make this product: $S=R\times\mathbb{Z}$ And give it ring structure with the operations. Sum by components and product:
$(a,n)(b,m)=(ab+ma+nb,nm)$
I've proven that this is a ring, it has unity, is commutative, and may have zero divisors, even if $R$ doesn't (At least in the case $R$ is an integral domain it's very easy to find the zero divisors).
I want to prove that $I=\{s\in S \mid sa=0,\forall a\in R\}$ is a prime ideal of $S$. I've already proved it's an ideal, but I don't know how to prove it is prime. How could I do that?
I've supposed $uv\in I$ then $\forall a\in R$, $(uv)a=0$, that is $u(va)=0$ and as $S$ is commutative $v(ua)=0$...